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610                    ON  THE PROBLEM OF RANDOM  VIBRATIONS,  AND                 [441
Two Dimensions.
If there is but one stretch of length I, the only possible value of r is of course I.
When there are two stretches of lengths ^ and la, r may vary from 12 - ^ to k + k, and then if 6 be the angle between them
- r2 = Zx2 + /22 - 2^ cos 0,  ........................ (17)
and                                      Biii0dd = rdr/I1l2 ............................ (18)
Since all angles 0 between 0 and TT are deemed equally probable, the chance of an angle between 6 and 6 + cW is ddj-jr. Accordingly the chance that the resultant r lies between r and r + dr is
rdr              ........................... (19)
Trlib sin 6 ' ...........................
or if with Prof. Pearson* we refer the probability to unit of area in the plane of representation,
fc^S^sin*
-
7
02 (^2) dA denoting the chance of the representative point lying in a small area dA at distance r from the origin.
If the stretches ^ and 12 are equal, (20) reduces to
Prof. Pearson's expression, applicable when r < 2.    When r > 2/1, <jf>2 O"2) = 0.
When there are three equal stretches (n = 3), <3 (r2) is expressible by elliptic functions f with a discontinuity in form as r passes through /.
For values of n from 4 to 7 inclusive, Pearson's work is founded upon the general functional relation J
(22) Putting r = 0, he deduces the special conclusion that
as is indeed evident a priori.
* Drapers'1 Company Research Memoirs, Biometric Series III., London, 1906. f Pearson (loc. cit.) attributes this evaluation to G. T. Bennett. J Compare Theory of Sound,  42 a.alwri.swt- of coii.set'utive i.solated pointH JH i!/, ;->' lhat. ii'(/,r be a large multiple of/, we may take